A046758 -id:A046758 - cp's OEIS frontend

A073048 Starting term of the first length-n chain of equidigital numbers (A046758).

5, 10, 1, 191, 13, 1134, 157, 1169312, 100149391, 1016258233, 107375450119, 10271593487494

Offset: 1

Lekraj Beedassy, Aug 31 2002

			13 is in the sequence because prime decompositions of 13 through 17, i.e. 13, 14=2*7, 15=3*5, 16=2^4, 17, use as many digits as their normal decimal representation.

Journal of Recreational Mathematics, 29(2) 138 1998, Solution to Problem 2209 "Equidigital Representation"

Cf. A046758.

a(9) corrected and a(11) from Donovan Johnson, Jul 30 2010

a(12) from Giovanni Resta, Aug 14 2013

A055642 Number of digits in the decimal expansion of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Henry Bottomley, Jun 06 2000

From Hieronymus Fischer, Jun 08 2012: (Start)
For n > 0 the first differences of A117804.
The total number of digits necessary to write down all the numbers 0, 1, 2, ..., n is A117804(n+1). (End)
Here a(0) = 1, but a different common convention is to consider that the expansion of 0 in any base b > 0 has 0 terms and digits. - M. F. Hasler, Dec 07 2018

			Examples:
999: 1 + floor(log_10(999)) = 1 + floor(2.x) = 1 + 2 = 3 or
      ceiling(log_10(999+1)) = ceiling(log_10(1000)) = ceiling(3) = 3;
1000: 1 + floor(log_10(1000)) = 1 + floor(3) = 1 + 3 = 4 or
      ceiling(log_10(1000+1)) = ceiling(log_10(1001)) = ceiling(3.x) = 4;
1001: 1 + floor(log_10(1001)) = 1 + floor(3.x) = 1 + 3 = 4 or
      ceiling(log_10(1001+1)) = ceiling(log_10(1002)) = ceiling(3.x) = 4;

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A043537, A178788, A046034, A019546, A054899, A122840, A055640, A055641, A102669-A102685, A117804, A160093, A160094, A196563, A196564, A000120, A000788, A023416, A059015 (for base 2).
Cf. A262190, A262198.
Cf. A007953: sum of digits.

a055642 :: Integer -> Int
a055642 = length . show  -- Reinhard Zumkeller, Feb 19 2012, Apr 26 2011

[ #Intseq(n): n in [0..105] ];   //  Bruno Berselli, Jun 30 2011
(Common Lisp) (defun A055642 (n) (if (zerop n) 1 (floor (log n 10)))) ; James Spahlinger, Oct 13 2012

A055642 := proc(n)
        max(1,ilog10(n)+1) ;
end proc:  # R. J. Mathar, Nov 30 2011

Join[{1}, Array[ Floor[ Log[10, 10# ]] &, 104]] (* Robert G. Wilson v, Jan 04 2006 *)
Join[{1},Table[IntegerLength[n],{n,104}]]
IntegerLength[Range[0,120]] (* Harvey P. Dale, Jul 02 2016 *)

a(n)=#Str(n) \\ M. F. Hasler, Nov 17 2008

A055642(n)=logint(n+!n,10)+1 \\ Increasingly faster than the above, for larger n. (About twice as fast for n ~ 10^7.) - M. F. Hasler, Dec 07 2018

def a(n): return len(str(n))
print([a(n) for n in range(121)]) # Michael S. Branicky, May 10 2022

def A055642(n): # Faster than len(str(n)) from ~ 50 digits on
    L = math.log10(n or 1)
    if L.is_integer() and 10**int(L)>n: return int(L or 1)
    return int(L)+1  # M. F. Hasler, Apr 08 2024

a(A046760(n)) < A050252(A046760(n)); a(A046759(n)) > A050252(A046759(n)). - Reinhard Zumkeller, Jun 21 2011
a(n) = A196563(n) + A196564(n).
a(n) = 1 + floor(log_10(n)) = 1 + A004216(n) = ceiling(log_10(n+1)) = A004218(n+1), if n >= 1. - Daniel Forgues, Mar 27 2014
a(A046758(n)) = A050252(A046758(n)). - Reinhard Zumkeller, Jun 21 2011
a(n) = A117804(n+1) - A117804(n), n > 0. - Hieronymus Fischer, Jun 08 2012
G.f.: g(x) = 1 + (1/(1-x))*Sum_{j>=0} x^(10^j). - Hieronymus Fischer, Jun 08 2012
a(n) = A262190(n) for n < 100; a(A262198(n)) != A262190(A262198(n)). - Reinhard Zumkeller, Sep 14 2015

A046759 Economical numbers: write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n) < l(n).

Original entry on oeis.org

125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250, 1280, 1331, 1369, 1458, 1536, 1681, 1701, 1715, 1792, 1849, 1875, 2048, 2187, 2197, 2209, 2401, 2560, 2809, 3125, 3481, 3584, 3645, 3721, 4096, 4374, 4375, 4489, 4802, 4913

Offset: 1

N. J. A. Sloane, Dec 11 1999

A050252(a(n)) < A055642(a(n)). - Reinhard Zumkeller, Jun 21 2011

			125 = 5^3, l(n) = 3 and D(n) = 2, so 125 is a member of the sequence.

Bernardo Recamán, The Bogota Puzzles, Courier Dover Publications, Inc., 2020, p. 77.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Glossary, economical number.
J.-M. De Koninck and F. Luca, On strings of consecutive economical numbers of arbitrary length, INTEGERS (2005), Volume: 5, Issue: 2, #A5.
Jean-Paul Delahaye, Les chasseurs de nombres premiers, Pour la Science, No. 258 April 1999. [v1].
R. G. E. Pinch, Economical numbers.
R. G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
W. Schneider, Economical Numbers.
G. Villemin's Almanach of Numbers, Nombres Economes.
Eric Weisstein's World of Mathematics, Economical Number.
Wikipedia, Frugal number.

Cf. A046758, A046760.

a046759 n = a046759_list !! (n-1)
a046759_list = filter (\n -> a050252 n < a055642 n) [1..]
-- Reinhard Zumkeller, Jun 21 2011

ecoQ[n_] := Total[ Length /@ IntegerDigits /@ Flatten[ FactorInteger[n] /. {p_, 1} -> p]] < Length[ IntegerDigits[n]]; Select[ Range[5000], ecoQ] (* Jean-François Alcover, Jul 28 2011 *)

is(n)=my(f=factor(n));sum(i=1,#f[,1], #Str(f[i,1])+if(f[i,2]>1, #Str(f[i,2])))<#Str(n) && n>1 \\ Charles R Greathouse IV, Feb 01 2013

from sympy import factorint
def ok(n): return n > 1 and sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items()) < len(str(n))
print([k for k in range(5000) if ok(k)]) # Michael S. Branicky, Dec 22 2024

More terms from Eric W. Weisstein

A046760 Wasteful numbers.

Original entry on oeis.org

4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 108, 110, 114

Offset: 1

N. J. A. Sloane

Write n as product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n)>l(n).

A050252(a(n)) > A055642(a(n)). - Reinhard Zumkeller, Jun 21 2011

			For n = 125 = 5^3, l(n) = 3 but D(n) = 2. So 125 is not a term of this sequence. [clarified by _Derek Orr_, Jan 30 2015]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Delahaye, Les chasseurs de nombres premiers.
R. G. E. Pinch, Economical numbers., arXiv:math/9802046 [math.NT], 1998.
Eric Weisstein's World of Mathematics, Wasteful Number.
Wikipedia, Extravagant number.

Cf. A046758, A046759.

a046760 n = a046760_list !! (n-1)
a046760_list = filter (\n -> a050252 n > a055642 n) [1..]
-- Reinhard Zumkeller, Aug 02 2013

Cases[Range[115], n_ /; Length[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]] > Length[IntegerDigits[n]]] (* Jean-François Alcover, Mar 21 2011 *)

for(n=1,100,s="";F=factor(n);for(i=1,#F[,1],s=concat(s,Str(F[i,1]));s=concat(s,Str(F[i,2])));c=0;for(j=1,#F[,2],if(F[j,2]==1,c++));if(#digits(n)<#s-c,print1(n,", "))) \\ Derek Orr, Jan 30 2015

from itertools import count, islice
from sympy import factorint
def A046760_gen(): # generator of terms
    return (n for n in count(1) if len(str(n)) < sum(len(str(p))+(len(str(e)) if e > 1 else 0) for p, e in factorint(n).items()))
A046760_list = list(islice(A046760_gen(),20)) # Chai Wah Wu, Feb 18 2022

A050252 Number of digits in the prime factorization of n (counting terms of the form p^1 as p).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 4, 3, 3, 2, 4, 2, 3, 3, 4, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 4, 2, 4, 3, 4, 3, 3, 3, 3, 2, 3, 4, 4, 3, 4

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.

Cf. A046758, A073048, A055642, A020639, A027748, A124010, A110475.

Cf. also A377369 (analog for base 2).

a050252 1 = 1
a050252 n = sum $ map a055642 $
            (a027748_row n) ++ (filter (> 1) $ a124010_row n)
-- Reinhard Zumkeller, Aug 03 2013, Jun 21 2011

nd[n_]:=Total@IntegerLength@Select[Flatten@FactorInteger[n],#>1&];Table[If[n==1,1,nd[n]],{n,102}] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

from sympy import factorint
def a(n): return 1 if n == 1 else sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items())
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Dec 27 2024

a(A192010(n)) = n and a(m) != n for m < A192010(n);

a(A046759(n)) < A055642(A046759(n)); a(A046758(n)) = A055642(A046758(n)); a(A046760(n)) > A055642(A046760(n)). [Reinhard Zumkeller, Jun 21 2011]

A254318 Hyper equidigital numbers.

Original entry on oeis.org

2, 3, 4, 5, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 28, 29, 31, 32, 35, 36, 37, 39, 41, 43, 46, 47, 49, 50, 53, 54, 58, 59, 61, 64, 67, 69, 71, 72, 73, 79, 81, 83, 89, 92, 93, 97, 98, 100, 101, 103, 104, 105, 106, 107, 109, 113, 116, 119

Offset: 1

Michel Lagneau, Jan 28 2015

The distinction between the equidigital numbers (A046758) is that only the distinct digits are counted instead of all digits. Hence the definition:

Write n as product of primes raised to powers, let D(n) = total number of distinct digits in product representation (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1) and nbd(n) = A043537(n) = number of distinct digits in n; sequence gives n such that D(n) = nbd(n).

			116 is in the sequence because 116 = 2^2*29 => D(116)= A043537(116)=2.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A043537, A046760, A046758, A046759, A254315, A254317, A254319, A254321.

Cases[Range[400], n_ /; Length[Union[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]]]==Length[Union[Flatten[IntegerDigits[n]]]]]

for(n=1,100,s=[];F=factor(n);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));if(#vecsort(digits(n),,8)==#vecsort(s,,8),print1(n,", "))) \\ Derek Orr, Jan 30 2015

A379373 Numbers k such that A050252(k) <= A055642(k).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 105, 106, 107, 109, 111, 112, 113, 115, 118, 119, 121, 122, 123, 125, 127, 128, 129, 131, 133, 134, 135

Offset: 1

Paolo Xausa, Dec 21 2024

Some authors call these numbers "economical numbers" (see links) and refer to A046759 as "frugal numbers", while other authors define "economical numbers" to be A046759.

First differs from A046758 at n = 54, where a(54) = 125 is missing from A046758.

			112 is a term because 112 = (2^4)*7; the total number of digits of (2, 4, 7) = 1 + 1 + 1 <= the number of digits of 112 (3).
125 is a term because 125 = 5^3; the total number of digits of (5, 3) = 1 + 1 <= the number of digits of 125 (3).

Paolo Xausa, Table of n, a(n) for n = 1..10000
Richard G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
Giovanni Resta, Economical numbers, Numbers Aplenty, 2013.

Union of A046758 and A046759.
Complement of A046760.
Cf. A050252, A055642.

A379373Q[k_] := Total[IntegerLength[Select[Flatten[FactorInteger[k]], # > 1 &]]] <= IntegerLength[k];
Select[Range[200], A379373Q]

from sympy import factorint
def ok(n): return n > 0 and sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items()) <= len(str(n))
print([k for k in range(136) if ok(k)]) # Michael S. Branicky, Dec 22 2024

A254319 Hyper economical numbers.

Original entry on oeis.org

27, 108, 125, 128, 129, 135, 138, 143, 159, 160, 184, 187, 189, 196, 207, 209, 216, 219, 243, 249, 256, 259, 265, 276, 295, 297, 329, 341, 351, 375, 403, 429, 451, 458, 469, 497, 512, 529, 621, 625, 671, 679, 729, 781, 795, 837, 841, 892, 896, 908, 916, 932

Offset: 1

Michel Lagneau, Jan 28 2015

The distinction between the economical numbers (A046759) is that the distinct digits are counted only instead all digits. Hence the definition:

Write n as product of primes raised to powers, let D(n) = total number of distinct digits in product representation (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1) and nbd(n) = A043537(n) number of distinct digits in n; sequence gives n such that D(n) < nbd(n).

			27 is in the sequence because 27 = 3 ^ 3 => D(27)= 1 < nbd(27)=2.

Cf. A043537, A046760, A046758, A046759, A254318, A254321.

Cases[Range[400], n_ /; Length[Union[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]]]< Length[Union[Flatten[IntegerDigits[n]]]]]

for(n=1,10^3,s=[];F=factor(n);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));if(#vecsort(digits(n),,8)>#vecsort(s,,8),print1(n,", "))) \\ Derek Orr, Jan 30 2015

A254321 Hyper Wasteful numbers.

Original entry on oeis.org

6, 8, 9, 22, 26, 30, 33, 34, 38, 40, 42, 44, 45, 48, 51, 52, 55, 56, 57, 60, 62, 63, 65, 66, 68, 70, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 94, 95, 96, 99, 102, 110, 111, 112, 114, 115, 117, 118, 122, 130, 133, 136, 141, 144, 152, 153, 155, 161

Offset: 1

Michel Lagneau, Jan 28 2015

The distinction between the Wasteful numbers (A046760) is that the distinct digits are counted only instead all digits. Hence the definition:

Write n as product of primes raised to powers, let D(n) = total number of distinct digits in product representation (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1) and nbd(n) = A043537(n) = number of distinct digits in n; sequence gives n such that D(n) > nbd(n).

			88 is in the sequence because 88 = 2 ^ 3 * 11 => D(88)=3 > nbd(88)=1.

Cf. A043537, A046760, A046758, A046759, A254318, A254319.

Cases[Range[400], n_ /; Length[Union[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]]] > Length[IntegerDigits[n]]]

for(n=1,100,s=[];F=factor(n);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));if(#vecsort(digits(n),,8)<#vecsort(s,,8),print1(n,", "))) \\ Derek Orr, Jan 30 2015

A172460 Partial sums of economical numbers A046759.

Original entry on oeis.org

125, 253, 496, 752, 1095, 1607, 2232, 2961, 3985, 5014, 6229, 7479, 8759, 10090, 11459, 12917, 14453, 16134, 17835, 19550, 21342, 23191, 25066, 27114, 29301, 31498, 33707, 36108, 38668, 41477, 44602, 48083, 51667, 55312, 59033, 63129

Offset: 1

Jonathan Vos Post, Feb 03 2010

The subsequence of prime partial sum of economical numbers begins: 1607, 6229, 12917, 76367. The subsequence of economical partial sum of economical numbers begins: 125; what is the first nontrivial economical partial sum of economical numbers?

			a(41) = 125 + 128 + 243 + 256 + 343 + 512 + 625 + 729 + 1024 + 1029 + 1215 + 1250 + 1280 + 1331 + 1369 + 1458 + 1536 + 1681 + 1701 + 1715 + 1792 + 1849 + 1875 + 2048 + 2187 + 2197 + 2209 + 2401 + 2560 + 2809 + 3125 + 3481 + 3584 + 3645 + 3721 + 4096 + 4374 + 4375 + 4489 + 4802 + 4913.

Cf. A000961, A046758, A046759, A046760.

a(n) = SUM[i=1..n] {n written as a product of primes raised to powers, where D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n)

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cp's OEIS Frontend

A073048 Starting term of the first length-n chain of equidigital numbers (A046758).

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A055642 Number of digits in the decimal expansion of n.

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A046759 Economical numbers: write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n) < l(n).

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A046760 Wasteful numbers.

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A050252 Number of digits in the prime factorization of n (counting terms of the form p^1 as p).

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A254318 Hyper equidigital numbers.

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A379373 Numbers k such that A050252(k) <= A055642(k).

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